Home » Blog » Determine whether ∑ 1 / n1 + 1/n converges

Determine whether ∑ 1 / n1 + 1/n converges

Test the following series for convergence:

    \[ \sum_{n=1}^{\infty} \frac{1}{n^{1 + \frac{1}{n}}}. \]


The series diverges.

Proof. First, we write

    \[ \frac{1}{n^{1+\frac{1}{n}}} = \frac{1}{n \cdot n^{\frac{1}{n}}}. \]

Then, we know n < 2^n for all n \geq 1. (We can deduce this from the Bernoulli inequality, (1+x)^n \geq 1 + nx with x = 1. We proved the Bernoulli inequality in this exercise, Section I.4.10, Exercise #14.) Therefore, n^{\frac{1}{n}} < 2 and we have

    \[ \frac{1}{n \cdot n^{\frac{1}{n}}} > \frac{1}{2n}. \]

Since the series \sum \frac{1}{2n} diverges we have established the divergence of the given series

    \[ \sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}. \qquad \blacksquare\]

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):